Functions: Notation, Domain and Range, Composite and Inverse Functions, Sketching Graphs

1. Function Notation

A function is a relation where each input (x-value) is associated with exactly one output (y-value). We use the notation $f(x)$ to represent a function.

Example: If $f(x) = 2x + 3$, then $f(4) = 2(4) + 3 = 11$. This means that when x = 4, the output is 11.

Function Rule: $y = f(x)$ represents the function where 'y' is the output and 'x' is the input.

Set Notation: A function can also be defined using set notation. For example, $f: A \rightarrow B$ means that the function 'f' maps elements from set 'A' to elements in set 'B'.

2. Domain and Range

2.1 Domain

The domain of a function is the set of all possible input values (x-values) for which the function is defined.

Example: For the function $f(x) = \sqrt{x - 2}$, the domain is $x \ge 2$ or $[2, \infty)$. We cannot take the square root of a negative number.

2.2 Range

The range of a function is the set of all possible output values (y-values) that the function can produce.

Example: For the function $f(x) = \sqrt{x - 2}$, and the domain is $x \ge 2$, the range is $y \ge 0$ or $[0, \infty)$. The square root of a non-negative number is always non-negative.

3. Composite Functions

A composite function is a function formed by applying one function to the output of another.

Notation: $(f \circ g)(x) = f(g(x))$

Example: Let $f(x) = x^2$ and $g(x) = x + 1$. Then $(f \circ g)(x) = f(g(x)) = f(x+1) = (x+1)^2$.

Important Note: The domain of the composite function is the set of all x-values in the domain of the inner function, $g(x)$.

4. Inverse Functions

An inverse function 'g' "undoes" the effect of the original function 'f'. If $f(a) = b$, then $g(b) = a$.

Notation: $g(x) = f^{-1}(x)$

Finding the Inverse:

1. Replace $f(x)$ with $y$.
2. Swap x and y.
3. Solve for y.
4. Replace y with $f^{-1}(x)$.

Example: Let $f(x) = 2x + 3$.

1. $y = 2x + 3$
2. $x = 2y + 3$
3. $x - 3 = 2y$ => $y = \frac{x - 3}{2}$
4. $f^{-1}(x) = \frac{x - 3}{2}$

Important Note: The domain of the inverse function is the range of the original function, and vice versa.

5. Sketching Graphs

To sketch the graph of a function, consider the following:

• Domain and Range: Determine the domain and range of the function.
• Key Points: Find any key points, such as x-intercepts (where $f(x) = 0$), y-intercepts (where $x = 0$), and points where the function has maximum or minimum values.
• Asymptotes: Identify any asymptotes (vertical, horizontal, or oblique).
• Shape: Consider the general shape of the function (e.g., a straight line, a parabola, a cubic).

Example: Sketching $y = x^2$

This is a parabola with its vertex at the origin (0, 0) and opens upwards. The domain is all real numbers, and the range is $y \ge 0$.

Topic	Description
Function Notation	$f(x)$ represents the output value for a given input value $x$.
Domain	The set of all possible input values (x-values).
Range	The set of all possible output values (y-values).
Composite Functions	$f(g(x)) = f(g(x))$ - applying one function to the output of another.
Inverse Functions	$f^{-1}(x)$ "undoes" the effect of $f(x)$.
Sketching Graphs	Consider domain, range, key points, asymptotes, and the general shape.